How to tackle a problem that is literally infinite? Just think – a lot

David Brooks

How do you do your job? This is how Tom Zhang does his:

“I just start to think about the question. I walk around the room, maybe outside,” said Zhang, a UNH mathematics professor who drew global attention last week for making a big step toward solving a number-theory problem that dates back to the ancient Greeks. “The most important thing is to keep thinking, for hours, for days, for weeks, for months, whenever you have time. ... Even when I’m sleeping.” ...
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“I just start to think about the question. I walk around the room, maybe outside,” said Zhang, a UNH mathematics professor who drew global attention last week for making a big step toward solving a number-theory problem that dates back to the ancient Greeks. “The most important thing is to keep thinking, for hours, for days, for weeks, for months, whenever you have time. ... Even when I’m sleeping.”

Zhang is best known in Durham as a calculus teacher – students give him rave reviews on RateMyProfessors.com, which is unusual for a mathematician with a strong accent – but in the mathematics community he’s now the twin-prime guy.

On May 13, Zhang (whose first name is Yitang, although in Durham he’s known as Tom) told a conference at Harvard University that he has proved a weak version of the twin-prime conjecture, one of the famous unsolved problems in mathematics.

A conjecture is an educated guess. In this case, the conjecture is that there are infinity pairs of prime numbers which are separated by exactly 2 – for example, 17 and 19, or 41 and 43, or 2,003,663,613 × 2^195,000 − 1 and 2,003,663,613 × 2^195,000 + 1.

(Primes, as you know, are numbers which can’t be divided evenly into any other whole number – that is, are divisible only by 1 and themselves. They are the basis of all mathematics, just as the periodic table is the basis of all chemistry.)

Mathematicians are pretty sure that no matter how far up the number line we wander, we will never run out of twin-prime pairs. But mathematicians don’t want pretty-sure-ness, they want proof.

Zhang has not, he is quick to point out, provided that final proof. What he has done is prove there are an infinity of primes which are separated by no more than 70 million.

This doesn’t sound very helpful but it’s actually a big step, because it moves the conjecture into the realm of the finite, where it’s easier to tackle. His step was big enough to draw article in prestigious places like the science journal Nature and plenty of attention from peers.

“I have got too many emails,” Zhang moaned as I interviewed him Tuesday morning, right after the publicity broke – in the midst of final exam week, a harried time for university professors.

At this point I would like to dazzle you by taking Zhang’s proof, which runs to 56 pages, and explaining it in a few pithy lines. Alas, while I got a bachelor’s degree in mathematics and have always been interested in number theory, that was long ago and far away, so I’m out of my depth.

Hinson, the department chair, sympathized with my inability. “Everybody can understand the statement, but the amount of mathematical firepower necessary to prove it is orders of magnitude greater,” he said.

Let’s stick with this summary from Zhang: “I found if we allowed the factorization of moduli ... then the former results can be improved.”

The reaction from the mathematical community seems positive, which isn’t always the case when a new proof of a famous long-standing problem is announced. We won’t be certain that it doesn’t contain some subtle error until other mathematicians gnaw on the final paper for a while, but there’s reason to be optimistic.

Zhang is a graduate of the University of Beijing who came to the U.S. for his doctorate (1992 from Purdue University) and has stayed.

He is my age – 57 – which is pretty gray-haired for a math breakthrough, usually something that comes from much younger people. In fact, the math world’s equivalent of the Nobel Prize, the Fields Medal, only goes to people under the age of 40.

As a historical note, Zhang was hired by Ken Appel, who was the subject of this column earlier last month following Appel’s death at age 78.

Appel is the best-known mathematician in UNH history, maybe New Hampshire history, because in the late 1970s he helped prove the four-color theorem, another famous and long-unsolved problem whose solution drew global attention.

Zhang’s work on the twin-prime conjecture doesn’t rise to the four-color-problem level of attention gathering, but it’s definitely of more interest to the outside world than most mathematical work. That is good news to Hinson, who like all mathematicians laments the world’s belief that his field is ossified.

“The most common question we get is this: Isn’t all of mathematics done?” he said. “The (mathematical) tools that people use are complete, they’ve been around a while, and it’s easy to confuse that with the absence of anything new.

“Something like this is refreshing, when people underestimate the depth of mathematics and the vitality of mathematical sciences practiced today. This is a window on mathematics research, shows that progress in mathematics is ongoing.”

That leads us to an obvious question: Who cares whether there are an infinity of twin-primes? What difference does it make?

It matters because mathematics is the language of the universe, and the more we understand it, the more chance we have of understanding anything. You can’t get a more basic level of mathematics – of science – than the properties of prime numbers. Knowing how to prove that an infinity of twin primes exists could help us understand something new and deep and useful.

“Mathematical models of reality are the mechanism for scientific theories. The deeper the understand of numbers themselves, the greater sophistication of models that can be produced,” said Hinson. “Number theory is at the heart, the core of mathematics, which is at the heart of science.”

Don’t believe that? Consider conic sections.

Ancient Greeks amused themselves by analyzing the properties of the curves made when you intersect a plane and a dunce’s cap (a cone). Fiddling with parabolas and hyperbolas and ovals was pure knowledge, beautiful and intriguing, but of no use to anyone.

Until 1,800 years later, when Kepler picked up that math and used it to create his laws about planetary orbits, shattering the ancient Earth-centered universe and helping launch the Renaissance and the modern world.

Without the “useless” theoretical math of conic sections, everything would be very, very different today.

The twin prime conjecture probably won’t have the same effect. But you never know until somebody tries – by thinking, and thinking, and thinking about it some more.

GraniteGeek appears Mondays in The Telegraph. David Brooks can be reached at 594-6531 or dbrooks@nashuatelegraph.com. Also, follow Brooks on Twitter (@granitegeek).